Study on establishment of st and ardized load spectrum on bogie frames of high?speed trains

Daoyun Chen · Qian Xiao · Minghui Mou · Shouguang Sun · Qiang Li

Abstract Establishing a structural load spectrum under actual operating conditions is a major problem in structural fatigue life analysis. This study introduces a load measuring method for the bogie frame structure. The quasi-static load-measuring frame can measure different load systems synchronously. The t test method is employed to evaluate the least test time to deduce the parent distribution. In order to fit the load spectrum distribution accurately, the kernel density estimation method is employed, which is based on the sample characteristics. The expansion factor method is used to deduce the maximum load.The formula for a st and ardized load spectrum is derived from the deduced maximum load, the linear factor between operating condition length and cumulative frequency, and the parent distribution of each load system. The damage consistency criterion is performed by solving the objective function with constraint conditions. The calibrated damage provides a suitable representation of the real damage under actual operating conditions. By processing and analyzing the load and stress spectral data of the measured lines, it is verified that the st and ardized load spectrum established in this paper is superior to the European specification and the Japanese specification in evaluating the fatigue reliability of the structure.

Keywords Load spectrum· Bogie frame· Kernel density· Expansion factor· Damage calibration

1 Introduction

Loading issues are critical to determine structural integrity and durability. The load spectrum has become an important way to express the loading condition of structures in many technical fields such as aviation [1–9], automotive engineering [10–13], off shore platforms [14–16] and mechanical engineering [17–19]. The bogie frame is a key load-bearing component for high-speed trains and has very complex loading conditions. In order to ensure a wide enough safety margin during the actual running of a high-speed train, the load spectrum of the bogie frame must be studied thoroughly.Currently, EN13749 [20], UIC615-4 [21] and JIS E 4207[22] are three types of load spectrum design specifications.The newly designed bogie frame is allowed to be installed in high-speed trains only after passing assessment of these load spectrum specifications. However, cracks can still be found in some bogie frames used in China. The re are two main reasons to explain this. An important reason is that these existing specifications are all established based on foreign testing railway lines. The operating conditions between foreign railway lines and Chinese railway lines has a large discrepancy so that these specifications can hardly meet train running conditions in China. Another reason is that the loads in these specifications are defined by the dynamic load coefficient method [23, 24], that is to say, the bogie frame loads are approximately determined by multiplying the load factor and static load. Although this method is easy to apply, the error level can also be very high compared with the actual conditions. The related parameters are usually determined in a conservative way, which shows certain limitations with the continuous improvement of the performance requirements for the bogie frame structure of high-speed trains.

Considering the actual railway conditions in China, the load spectrum under typical operating conditions must be established as soon as possible to avoid potential fatigue failure in bogie frame structures. The typical operating conditions are those that form the main part of the railway line.Generally speaking, the typical operating conditions include a high-speed straight line, low-speed straight line, high-speed curve, low-speed curve, tunnel, straight passage through a switch and side passage through a switch. The load spectrum under typical operating conditions should be able to predict accurate load spectral values such as load amplitude and load frequencies when the railway line length of each operating condition is provided. It is clear that the load spectrum under typical operating conditions should be parameterized. The only unknown parameter is the railway line length of operating condition and the other known parameters are from statistics of measured data. The load spectrum in this study can be called a standardized load spectrum because all the load spectra of the test train can be calculated by the established load spectrum formula when the railway line length of each operating condition is known. This saves a lot of manpower and material resources, as well as improves the accuracy and efficiency of the evaluation results.

The research provides the whole establishment process of standardized load spectrum on the bogie frames of highspeed trains, which includes the following sections. Section2 discusses the distribution fitting principle of load spectrum. In Sect.3, the extreme value inference principle of the load spectrum is presented. Section4 shows how to determine the least test times of high-confidence load spectra. Section5 provides the formula derivation process of st and ardized load spectrum. The formula is able to evaluate the specific load amplitude and load frequency when railway line length of each operating condition is provided. The calibration of damage consistency is applied based on the artificial fish-swarm algorithm in Sect.6 to get an accurate st and ardized load spectrum.

Fig. 1 Load system of the bogie frame

2 Load system s and actual running test

The bogie frame is a typical welded steel box structure, as shown in Fig.1. It consists of two side beams, two cross beams, and two longitudinal beams. The side beams are fishbelly shaped with a concave middle section. The primary vertical damper brackets, localization supports and braking hanger brackets are welded onto the side beams. The bogie frame is made of carbon steel S355J2, and its material composition and some other mechanical property parameters are listed in Table1. Structurally, the bogie frame of a highspeed train is a typical, complex frame structure. In terms of bearing condition, the bogie frame bears approximately 20 independent loads, which is a typical complex multisource load system. According to the movement characteristics and design criteria of bogie frames, the load systems are composed of vertical, rolling, torsional, lateral, gearbox and braking load systems. Generally speaking, the vertical and rolling load systems are generated by vertical load change and rolling movements of the car body. The torsional load system is based on the inequality of the four axle-box counter forces caused by deficiencies in railway line tracks under vertical static loads [27]. The lateral load system derives from the lateral vibration of the bearing quality and installation components. The gearbox load system is from the gearbox vibration. The braking load system is generated by forces caused by vertical vibration of braking hanger brackets.

The forces of each load system are measured based on quasi-static theory. Each load is directly measured from one set of strain-bridge paths. Before carrying out the formal line load test, the frame is calibrated in the laboratory.The hydraulic actuator is used to apply a certain amount of static load to the corresponding load system of the frame,the strain values at the load identification points of each load system are recorded, and the transfer coefficient between the load input and the strain response is obtained. When the measured strain time history of the line is obtained, the load time history of each load system can be obtained by multiplying with the transfer coefficient. The calibration test for the load-measuring frame was operated through the multichannel loading calibrating experimental bench, which is shown in Fig.2. The multistage loading tests under quasistatic load systems were conducted on the frame and the output voltage of the load measuring systems was the n collected at the same time. Finally, the load-strain transfer coefficients of each load system were calculated.

After demarcation, the load-measuring frame was installed on CR400BF high-speed train. The running tests under different operational conditions were conducted on the Datong–Xi’an dedicated passenger line. The electronic data acquisition (eDAQ) multichannel data acquisition system with a sampling frequency of 500Hz was used for data acquisition. The measured data were the n separated according to the operating conditions by data processing software. The load spectra of each load system under seven different operating conditions were the n obtained as sample load spectra. In order to judge whether the sample load spectra were enough to infer the parent distributions, a piece of MATLAB code was compiled based on t test. Figure3 shows the detailed execution process for evaluating the least test time.

Table 1 Material composition and mechanical property parameters of S355J2 [25, 26]

Fig. 2 Calibration test for the load-measuring frame

After operating the two independent samples t test by MATLAB, the critical sampling points of each load system under each operating condition are obtained. Figure4 shows the statistics of the least measurement number under highspeed straight line operating condition. It is clear that the biggest critical sampling points are 322 points on the gearbox load system which means the least measurement time under high-speed straight line operating condition is 322.

3 Establishment of standardized load spectrum

3.1 Distribution fitting principle of load spectrum

The kernel density function [28] is applied to fit the load spectrum distribution, which is a nonparametric estimation method. When the kernel density estimation method is used,a priori know ledge of data distribution is not needed. As a result, no assumption is added to data distribution. This method is used to study the characteristics of data distribution from the sample itself, which means strong adaptability.When calculating the data distribution around a point by kernel density function, the statistical characteristics are fully considered. We assume the probability density function and distribution function of the random variable x as f(x) and F(x), respectively, the n

Fig. 3 Execution process for evaluating the least test time

where h represents group gap, and P(x ? h < X < x + h) represents the proportion of samples falling into the interval[x ? h, x + h].

Fig. 4 Statistics of the least measurement number under high-speed straight line operating condition. a Vertical load spectrum, b rolling load spectrum, c torsional load spectrum, d lateral load spectrum, e gearbox load spectrum, f braking load spectrum

According to Eq.(1), the probability density estimation can also be written as

where N(x ? h < X < x + h) represents the number that X1,X2,…, Xnfalling into [x ? h, x + h].

The function f(x) is discontinuous at the boundary of each interval, so part of the probability information will be lost.For this reason, we define the following function

The probability density estimation can the n be rewritten as

If the functioning(x) is replaced by kernel function, the n the kernel density estimation at point x can be rewritten as

where K(·) is the kernel function, h represents b and width coefficient, and n is the sample size.

The accuracy of the kernel density estimation depends on the selection of the kernel function and the b and width coefficient. When the kernel function is fixed, the greater the b and width, the smoother the estimated density function. However, some features of f(x) will be covered up. The smaller the b and width coefficient, the better the estimated density curve and the sample fitting. However, the variance increases. The effect of different kernel functions on the estimation error is very small. The kernel function must meet the following conditions

where sup|?|is the supremum of the function,

The commonly used kernel function includes a uniform function, trigonometric function, Epanechikov function,Gauss function, and cosine function. The kernel function in this paper is chosen as a Gauss function

The best b and width can be determined by minimizing asymptotic mean integrated square error (AM ISE). According to the probability density function, the deviation and variance can be calculated as where u = (x ? Xi)/h, O(·) is the high order of the Taylor expansion of f (x + uh) at x.

Table 2 Mathematical expectation of kernel density parameters under high-speed straight line operating condition

The mean square error (MSE) ofcan be derived from Eqs.(10) and (11)

Further integrating ingMSEto eliminate the influence of different values of X

Neglecting the high-order term of the MSE of the integral and preserving the main term, the AM ISE can the n be determined. For the AM ISE to reach the minimum b and width,the following equation should be satisfied

After derivation at h, the calculated best b and width is

According to the rule of thumb, assuming that f(x) conforms to a normal distribution family with the standard deviation of σ and taking the Gauss kernel function as the kernel function, the best b and width is finally shown as

After data processing and calculation based on the kernel density function, the mathematical expectation of kernel density parameters under different operating conditions can be obtained. Table2 shows the mathematical expectation of kernel density parameters under high-speed straight line operating conditions. Table3 lists the template load spectrum under high-speed straight line operating conditions.As can be seen from the template load spectrum, high-load frequencies with low amplitudes are dominant, which shows the excellent track quality of this line.

3.2 Extreme value inference principle of load spectrum

The maximum load level of the measured load spectrum corresponds to accumulative frequency. If we want to get the maximum value of the load spectrum according to the given accumulative frequency, the maximum load must be deduced. In this study, the expansion factor method [29] is employed to deduce the maximum load. The precondition of this method is to transform the actual sample load accumulative frequency into standardized accumulative distribution.The distribution function has the form

where X is the load level, Xmis the maximum load, H is the accumulative frequency related to X, H0is the maximum accumulative frequency, and n is the total group number of load spectrum.

The core of expansion factor method is to move and prolong the accumulative frequency curve. The first step is to determine the expansion factor of curve moving. In this way,curve 1 should be moved to curve 2 in Fig.5. The expansion factor can be calculated by γ = H02/H01. The second step is to determine the position and shape of curve 3 in Fig.5 so that curve 2 and curve 3 can transmit smoothly and form an entire standardized accumulative frequency curve.

The formula of curve 3 can be assumed as

where a is the unknown factor.

Equation(18) can be logically transformed as

with lg H3=and X3= Xm1, we can get

Table 3 Template load spectrum under high-speed straight line operating conditions

Fig. 5 Expansion factor method

where Xm1is the maximum load of accumulative frequency spectrum before expansion, H01is the maximum accumulative traversing frequency before expansion, Xm3is the maximum load of accumulative frequency spectrum after expansion, and H02is the maximum accumulative traversing frequency after expansion.

3.3 Formula derivation of standardized load spectrum

The derivation of the standardized load spectrum formula under typical operating conditions can be divided into the following steps. Firstly, we must establish a formula for calculating the cumulative frequency of the standardized load spectrum. The length of railway line under different operating conditions is Li, i = 1, 2,…, 7. This represents seven different operating conditions including high-speed straight line, low-speed straight line, high-speed curve, low-speed curve, tunnel, straight passage through switch, and side passage through switch. Nij, j = 1,2,…,6 represents six load types including vertical, rolling, torsional, lateral, gearbox, and braking. kijis the slope of the straight line with scattered points and has the following relation

Figure6 shows the dispersion linear fitting between railway line length of operating condition and accumulative frequency under high-speed straight line operating condition. It is clear that the scattered points of the measured data are more evenly distributed on both sides of the fitting line.The upper and lower limits of confidence intervals for 95%confidence are also shown in Fig.6. The width of the linear fit confidence interval of gearbox load system is the largest.The width of the linear fit confidence interval of torsional load system is the smallest. Table4 shows linear fitting coefficient values of typical operating conditions between operating condition length and accumulative frequency.

Secondly, we must estimate the maximum load of the standardized load spectrum and determine the load amplitude of each level of standard spectrum corresponding to it.The amplitude and frequency of the template load spectrum are Fijuand nijurespectively. u is the load spectrum level.The accumulative frequency of template load spectrum can be shown as. Next, we get the maximum load of standardized load spectrum according to the expansion factor method as

Each load amplitude level of standardized load spectrum is shown as

Finally, combining with probability density function of parent load spectrum, the frequency of standard load spectrum at all levels is deduced as

3.4 Equivalent load analysis

In order to quantify the measured load of each load system under typical operating conditions, the measured load of each load system is considered to be equivalent to the equivalent load of constant amplitude cycle. Assuming the strain produced by the load system Fiat its load identification point is ingi, the n we have

Fig. 6 Dispersion linear fitting between operating condition length and accumulative frequency under high-speed straight line operating condition. a Vertical load spectrum, b rolling load spectrum, c torsional load spectrum, d lateral load spectrum, e gearbox load spectrum, f braking load spectrum

Table 4 Linear fitting coefficient values of typical operating conditions between operating condition length and accumulative frequency

where ξiis the load-strain transfer coefficient of load system Fi.

According to the linear cumulative damage criterion and the S–N curve of the material, the damage of a stress spectrum block can be calculated as

where D1is the fatigue damage value of a stress spectral block, C1and m are parameters of S–N curve, niis the number of cycles corresponding to the amplitude of stress at each level, σiis the stress amplitude at each level, Niis the fatigue life corresponding to current stress amplitude, N is the number of levels of stress spectrum, and E is the elastic modulus of materials.

Assuming that the damage caused by N times the equivalent load of Feqvto the measuring point is D, the n we have

where N0is cycle times corresponding to fatigue strength of materials.

The fatigue damage caused by the measured stress spectrum of train mileage L1is assumed to be D1, and the fatigue damage caused by running mileage L under constant amplitude load is assumed to be D, the n we have

Finally, the following formula can be derived from Eqs.(29)–(31)

Equation(32) is the equivalent load calculation formula.The total running mileage of a high-speed train during its lifetime is 1.5 × 107km. In order to calculate the equivalent loads under typical operating conditions, the measured mileage of each typical operating condition is extended to 1.5 × 107km to obtain the equivalent load corresponding to the whole life of the frame under typical operating conditions.

Figure7 shows the final 100% stacked bar chart, showing the equivalent load size of each load system in percentage form, where OC is the operating condition. As shown in Fig.7, in all seven typical operating conditions, the equivalent load of the gearbox occupies the largest proportion in the load system of the operating condition, among which the equivalent load of the gearbox in the tunnel condition is the highest, reaching 53%, followed by the high-speed straight line operating condition with 49%. The proportion of the gearbox equivalent load of the high-speed curve and the low-speed curve is the same, which is 48%, the gearbox load of the straight passage through the switch takes up 47%, while the gearbox load proportion of the side passage through the switch is the least, which is only 25%. Except for the side passage through the switch operating condition, the braking equivalent load occupies the second largest percentage of the load system in the rest operating conditions. The ratio of the equivalent load of vertical, rolling and torsional load system is very small, which is between 3% and 7%. In the side passage through the switch operating condition, the equivalent loads of the lateral load system and rolling load system are 21% and 18%, respectively, and the proportion of equivalent load between the torsional load system and vertical load system is the smallest, which are 14% and 8%,respectively. However, it is higher than the equivalent load ratio of torsional load system and vertical load system under other operating conditions.

Fig. 7 Percentage of equivalent loads in typical load system

4 Calibration of damage consistency on st and ardized load spectrum

The re is always a certain difference between the damage calculated according to the measured load spectrum and the actual damage at the measurement point of the frame. However, reliability analysis of structures requires load spectrum energy to reflect the stress under the condition of consistent damage. Therefore, it is necessary to calibrate the damage consistency of the standardized load spectrum of the bogie frame. The bogie frame has a complex structure and many measurement points. In order to calibrate the damage consistency, it is necessary to select the points with large stress response, which are called damage sensitive points. When applying loads on the bogie frame, the stresses on damage sensitive points show a larger response than the other measuring points, which leads to higher damage compared with the other measuring points. According to the analysis of the finite element calculation and the measured data, the specific locations of damage sensitive points are shown in Fig.8.

The main basis for calculating damage are Miner’s linear cumulative damage rule [30] and the S–N curve. The actual damage can be calculated by

where n is the level of stress amplitude spectrum, l is the measured mileage, L is the mileage of safety operating, q is the number of damage sensitive points of frame corresponding to load spectrum, m is the constant of S–N curve, N is the stress cycle number corresponding to fatigue limit of welded joints, σois the fatigue allowable stress of welded joints, σpuis the u-order stress amplitude of the measured stress spectrum of the damage sensitive point p corresponding to the load spectrum, and Mpuis the u-order frequency of the measured stress spectrum of the damage sensitive point p corresponding to the load spectrum.

To calculate the damage caused by the load spectrum,it is necessary to overlay the damage caused by the load spectrum of each load system at the damage sensitive point.

We can calculate the damage of sensitive point corresponding to measured load spectrum as

where n is the load spectrum level, σpjuis the u-order stress response of damage sensitive point p corresponding to load spectrum of load system j on the frame, Mpjuis the u-order loading frequency corresponding to load system j on the damage sensitive point p, φpjis the load-strain transfer coefficient of the damage sensitive point p corresponding to the load spectrum of the frame and load system j, and Fpjuis the amplitude of u-order load corresponding to the load system j.

The necessary condition of the frame load spectrum suitable for reliability test evaluation and design is that the damage of sensitive point corresponding to the frame load spectrum cannot be smaller than the actual damage of the frame under the condition of service. Therefore, the calibration coefficient method is used to calibrate the measured load spectrum. The optimization equation is established to calculate the calibration coefficients based on damage consistency criterion and genetic algorithm. The object function and the constraint condition can be described as

where γijis the calibration coefficient of load system j under typical operating condition i.

Table5 shows the load calibration coefficients of different load systems under different operating conditions. The load calibration coefficients are necessary parameters of the standardized load spectrum.

5 Superiority verification o f the standardized load spectrum

The standardized load spectrum, which is established by statistical analysis and other means, will be applied to the structural reliability evaluation of the bogie frame. In order to evaluate the superiority of the standardized load spectrum for the reliability evaluation of the frame structure, the precision of standardized load spectrum will be researched.In this section, the precision of standardized load spectrum is studied based on linear cumulative damage. Firstly, the railway line length of typical operating conditions of a given measured line is counted and the standardized load spectrum of each typical condition is calculated. The n the standardized load spectrum under each typical operating condition is transformed into the equivalent load spectrum of each load system. Finally, the total cumulative frequency of the equivalent load spectrum of standardized load spectrum, the measured load spectrum and the measured stress spectrum at the key fatigue measurement points are extended to the specified cycle times. The damage caused by each spectrum is calculated and compared with the European standard and the Japanese standard.

Figure9 shows the Shenyangbei–Dalianbei railway section of Harbin–Dalian high-speed railway, which includes a high-speed straight line operating condition, low-speed straight line operating condition, high-speed curve operating condition, low-speed curve operating condition, tunnel operating condition, straight passage through a switch and side passage through a switch.

Fig. 8 Map of the location of damage sensitive points. a View 1, b view 2, c view 3, d view 4, e view 5, f view 6, g view 7

Fig. 8 (continued)

The selected test lines are identified and separated, and the railway line length information of each typical operating condition is shown in Table6. It can be seen from Table6 that the main operating conditions of the selected test lines are high-speed straight lines and high-speed curves, while low-speed curves, low-speed straight lines, and straight passage through a switch take the second place. The ratio of the railway line length of the side passage through a switch and tunnel operating condition is small with 1613m and 708m respectively.

Table 6 Condition length statistics of test line (unit: m)

The standardized load spectrum of each load system under typical working conditions is obtained by inserting the parameters of actual railway line length and other parameters into the standardized load spectrum formula, and the measured load spectrum is obtained by rain flow counting the measured load data.

For each stress measurement point of the frame, a section of the measured stress spectrum can be obtained by data processing according to the measured strain time history of the strain gauge attached to it. However, for the standardized load spectrum and the measured load spectrum, the stress spectrum at the stress measurement point can be obtained by the transfer coefficient between the load and the stress measurement point. Different from the measured stress spectrum at the stress measurement point, the stress spectrum at the stress measurement point calculated by the standardized load spectrum and the measured load spectrum is no longer just one section of the spectrum. The load spectrum is related not only to the operating condition, but also to the load system, so the stress spectrum at the measurement point is also related to operating condition and load system, which is not conducive to the subsequent comparison with the measured stress spectrum. According to the basic compilation criterion of the load spectrum, the stress spectrum produced by the load spectrum of different load system under different operating conditions at the measurement point is fused into one spectrum, that is, the equivalent spectrum. The steps are as follows.

Firstly, the amplitude of the equivalent spectrum should be compiled. The calculation principle of the amplitude of the equivalent spectrum is shown in Fig.10.

For each stress spectrum, it is necessary to find out the maximum value of the amplitude of each stress spectrum AMmaxand the amplitude interval G of the stress spectrum related to the maximum value. In this paper, according to the equivalent principle of 16-level load spectrum, the maximum load value of the equivalent load spectrum is AMmax+ G/2. The amplitude of the load spectrum is usually represented by the median value of the load in each group,so the amplitude of the equivalent load spectrum per stage can be defined as

Table 5 Load calibration coefficients

Fig. 9 Shenyangbei–Dalianbei railway section of Harbin–Dalian high-speed railway

Secondly, the frequency of falling into each interval of equivalent spectrum must be counted. For the determined equivalent spectrum interval, the frequency corresponding to the stress amplitude of each stress spectrum, which belongs to the interval of the equivalent spectrum, is counted into the frequency of the equivalent spectrum of that stage and so on until all the frequencies of the stress spectrum fall into the corresponding equivalent spectrum interval.

Fig. 10 Principle of amplitude grouping of equivalent load spectrum

Fig. 11 Comparison with the true equivalent stress ratio

In the fatigue reliability assessment, the international specification frequency of dynamic load is consistent with the corresponding frequency of dynamic load in the whole life cycle of the structure, while the frequency of equivalent spectrum and measured stress spectrum is obviously only a part of the frequency in the whole life cycle. In order to compare all types of spectrum with international specifications,it is necessary to exp and the cumulative frequency of the spectrum. The expansion of a spectrum depends on its probability density distribution. In this paper, we use the kernel density function to describe the probability density distribution of the spectrum. The load frequency of each spectrum can be obtained by calculating the probability value corresponding to the load amplitude of the spectrum and multiplying the number of load cycles in the whole period.

Equivalent stress is one way to evaluate the fatigue reliability of structure. The derivation process of equivalent stress formula is similar to that of equivalent load formula.We calculate and compare the ratio between real equivalent stress and measured load spectrum, standardized load spectrum, the European specification, and the Japanese specification at each stress measurement point of the frame,which is shown in Fig.11. In Fig.11, A represents gearbox suspension seat, B represents connecting part of traction rod and crossbeam, C represents transverse shock absorber seat,D represents positioning arm seat, E represents connection part of lateral beam and side beam, F represents connection part of lateral beam and longitudinal beam, G represents brake suspension seat, H represents motor suspension seat,I represents snake-resistant shock absorber seat.

It can be seen from Fig.11 that the trend of the ratio between the stress produced by the standardized load spectrum and the true equivalent stress at each measurement point is basically consistent with that of the measured load spectrum as far as the overall trend of the scattered point curve is concerned. At some measurement points of the connecting part of the traction rod and the crossbeam, the connection part of the transverse side beam, the connection part of the transverse longitudinal beam and the anti-snake damper seat, the ratio between the real equivalent stress and the stress produced by the standardized load spectrum and the measured load spectrum has a local peak value. The maximum value of the ratio between the standardized load spectrum stress and the real equivalent stress is 6.1, and the maximum value of the ratio between the measured load spectrum stress and the real equivalent stress is 4.9. The ratio between true equivalent stress and the stress produced by measured load spectrum, the European specification and the Japanese specification at some measurement points is less than 1, and the ratio between the true equivalent stress and the stress produced by the standardized load spectrum at all the measurement points is greater than 1. It is shown that the standardized load spectrum can achieve the equivalent stress coverage of all the measurement points of the frame.

The ratio between the standardized load spectrum stress and the true equivalent stress of some measurement points is smaller than that of the European specification and the Japanese specification, which indicates that the accuracy of the standardized load spectrum assessment at some sites is higher than that of the European specification and the Japanese specification. Since the standardized load spectrum stress at all the measurement points is greater than the true equivalent stress, the standardized load spectrum is conservative and safe for the strength evaluation of bogie frame.

6 Conclusions

The establishment of standardized load spectrum on bogie frames of high-speed trains under typical operating conditions is performed in this study. The following conclusions are drawn.

1. The t test method is effective to evaluate the least test time to ensure that the statistical result from the sample load spectra is able to reflect the parent distribution characteristics.

2. The kernel density estimation method can describe the load spectrum distribution effectively. The most important step of kernel density estimation is to calculate the b and width factor. The expansion factor method can be used to deduce the maximum load precisely.

3. The formula for the standardized load spectrum derives from the deduced maximum load, the linear factor between operating condition length and cumulative frequency and the parent distribution of each load system.The only unknown parameter of the formula is operating condition length. The other known parameters are obtained from the statistics of the sample load spectra.

4. By employing equivalent load analysis, the equivalent load of each load system under each operating condition can be clearly expressed.

5. The damage consistency criterion can be performed by solving the objective function with constraint conditions. The calibrated damage provides a suitable representation of the real damage under actual operating conditions.

6. The superiority of the standardized load spectrum compared with European specification and Japanese specification is verified by calculating the equivalent stress.Compared with European specification and Japanese specification, the trend of the equivalent stress ratio produced by the standardized load spectrum at the measurement point is the closest to that of the true equivalent stress ratio, and the stress generated by the standardized load spectrum at all the measurement points is larger than the true equivalent stress. It is shown that the standardized load spectrum can achieve the equivalent stress coverage of all the measurement points of the frame.

AcknowledgementsThis work was supported by the National Natural Science Foundation of China (Grant 51565013).